Nominal Versus Actual Spatial Resolution: Comparison of Directivity and Frequency-Dependent Effective Sensitive Element Size for Membrane, Needle, Capsule, and Fiber-Optic Hydrophones

Frequency-dependent effective sensitive element radius$\boldsymbol{a}_{\text {eff}}{({f})}$is a key parameter for elucidating physical mechanisms of hydrophone operation. In addition, it is essential to know$\boldsymbol{a}_{\text {eff}}{({f})}$to correct for hydrophone output voltage reduction due to spatial averaging across the hydrophone sensitive element surface. At low frequencies,$\boldsymbol{a}_{\text {eff}}{({f})}$is greater than geometrical sensitive element radius$\boldsymbol{a}_{g}$. Consequently, at low frequencies, investigators can overrate their hydrophone spatial resolution. Empirical models for$\boldsymbol{a}_{\text {eff}}{({f})}$for membrane, needle, and fiber-optic hydrophones have been obtained previously. In this article, an empirical model for$\boldsymbol{a}_{\text {eff}}{({f})}$for capsule hydrophones is presented, so that models are now available for the four most common hydrophone types used in biomedical ultrasound. The$\boldsymbol{a}_{\text {eff}}{({f})}$value was estimated from directivity measurements (over the range from 1 to 20 MHz) for five capsule hydrophones (three with${d}_{g}= {2} \boldsymbol{a}_{g}= {85}\,\,\mu \text{m}$and two with${d}_{g}= {2} \boldsymbol{a}_{g}= {200}\,\,\mu \text{m}$). The results suggest that capsule hydrophones behave according to a “rigid piston” model for${k}~{\boldsymbol {a}}~_{g} \ge {0.7}$(${k} = {2}\pi $/wavelength). Comparing the four hydrophone types, the low-frequency discrepancy between$\boldsymbol{a}_{\text {eff}}{({f})}$and$\boldsymbol{a}_{g}$was found to be greatest for membrane hydrophones, followed by capsule hydrophones, and smallest for needle and fiber-optic hydrophones. Empirical models for$\boldsymbol{a}_{\text {eff}}{({f})}$are helpful for choosing an appropriate hydrophone for an experiment and for correcting for spatial averaging (over the sensitive element surface) in pressure and beamwidth measurements. When reporting hydrophone-based pressure measurements, investigators should specify$\boldsymbol{a}_{\text {eff}}$at the center frequency (which may be estimated from the models presented here) in addition to$\boldsymbol{a}_{g}$.